In this manuscript two BMO estimates are obtained, one for Linear Elasticity and one for Nonlinear Elasticity. It is first shown that the BMO-seminorm of the gradient of a vector-valued mapping is bounded above by a constant times the BMO-seminorm of the symmetric part of its gradient, that is, a Korn inequality in BMO. The uniqueness of equilibrium for a finite deformation whose principal stresses are everywhere nonnegative is then considered. It is shown that when the second variation of the energy, when considered as a function of the strain, is uniformly positive definite at such an equilibrium solution, then there is a BMO-neighborhood in strain space where there are no other equilibrium solutions.

中文翻译：

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BMO 和弹性：Korn 不等式；张力的局部独特性

在本手稿中，获得了两个 BMO 估计值，一个用于线性弹性，另一个用于非线性弹性。首先表明，向量值映射的梯度的 BMO-seminorm 的边界是其梯度对称部分的 BMO-seminorm 的常数倍，即 BMO 中的 Korn 不等式。然后考虑主应力处处非负的有限变形的平衡唯一性。结果表明，当能量的第二个变化作为应变的函数时，在这样的平衡解中是一致正定的，那么在没有其他平衡解的应变空间中存在 BMO 邻域。